Further, an analysis of the maximum transverse displacement (\(\tilde{w}_0\) ) [Fig. REF (c)] shows that an increase in the average porosity, for a fixed \(P_0\) , does not result in a consistent softening of the porous beam (that is, an increase in \(\tilde{w}_0\) ). This observation is in contrast to the expected behavior based on a classical IO model. However, an increase in the strength of nonlocality (e.g. via a decrease in the average VO [\(\alpha _0=\langle \alpha (x) \rangle \) ; \(\alpha _0\)  for beams with \(p_0=\lbrace 0,0.20,0.22,0.24\rbrace \)  are \( \alpha _0 = \lbrace 1,0.88,0.84,0.86\rbrace \) ]) correlates in a consistent manner with the softening phenomenon In constant-order fractional kinematic approaches to nonlocal elasticity, a decrease in the order results in an increase in the strength of the nonlocal effects (degree of nonlocality) and in a consistent softening behavior.. This result also holds true for \(\langle w_0 \rangle \)  and \(\overline{w}_0\) , and even for the thermoelastic response results presented in [1]}. Thus, it emerges that at the continuum level the average nonlocal strength \(\alpha _0\) , and not the average porosity \(p_0\) , provides a much more appropriate and physically consistent representation of the porous microstructure. On a more theoretical note, we emphasize that the softening behavior observed in the porous beams (in both the nonlinear elastic and thermoelastic response) was captured consistently (i.e. without any display of paradoxical effects [2]}, [3]}) by the VO theory. This result is possible thanks to the underlying foundation provided by the displacement-driven formulation [based on differ-integral kinematic relations, see Eq. (REF )], which is intrinsically well-posed and thermodynamically consistent [4]}.
